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Econometrica, Vol. 80, No. 6 (November, 2012), 2469–2509
CAPITAL MOBILITY AND ASSET PRICINGDARRELL DUFFIE
Graduate School of Business, Stanford University, Stanford, CA 94305, U.S.A.
BRUNO STRULOVICINorthwestern University, Evanston, IL 60208, U.S.A.

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Econometrica, Vol. 80, No. 6 (November, 2012), 2469–2509

CAPITAL MOBILITY AND ASSET PRICINGBY DARRELL DUFFIE AND BRUNO STRULOVICI1
We present a model for the equilibrium movement of capital between asset markets
that are distinguished only by the levels of capital invested in each. Investment in thatmarket with the greatest amount of capital earns the lowest risk premium. Intermediaries optimally trade off the costs of intermediation against fees that depend on the gain
they can offer to investors for moving their capital to the market with the higher mean
return. The bargaining power of an investor depends on potential access to alternativeintermediaries. In equilibrium, the speeds of adjustment of mean returns and of capital
between the two markets are increasing in the degree to which capital is imbalanced
between the two markets, and can be reduced by competition among intermediaries.
KEYWORDS: Capital mobility, market frictions, financial intermediation, law of one
price.

1. INTRODUCTION
WE PRESENT A MODEL for the equilibrium movement of capital between two
partially segmented markets that are identical except for the amounts of capital invested in each. Equilibrium conditional mean rates of return vary acrossthe markets according to the levels of capital invested in the respective markets. As a matter of supply and demand within each market, that market withthe greater amount of capital earns lower conditional mean returns. Given a
sufficient disparity in the capital levels in the markets, intermediaries find it
optimal to search for investors in the market with “surplus” capital and offer
them the opportunity to move their capital to the other market, which offers
higher risk premia. Intermediaries charge investors a fee that is based on theirgain from the move and based on the degree of competition in the market forintermediation. The equilibrium behavior of intermediaries is solved analytically, and characterized. Increased competition among intermediaries can insome cases reduce capital mobility.
In our equilibrium model, the greater the heterogeneity in capital levels
across the markets, the more intensive are intermediaries’ efforts to rebalance
the distribution of capital across the markets, and the greater is the rate ofconvergence of the mean rates of return of different assets toward a common
1
We are grateful for reactions at Oxford University, the Gerzensee European Summer Symposium in Financial Markets, the University of Toulouse, The London School of Economics,
London Business School, Yale University, NBER Asset Pricing Conference, Columbia University, Northwestern University, and especially for comments from Bruno Biais, Eddie Dekel, Julien
Hugonnier, Jean-Charles Rochet, Larry Samuelson, Avanidhar (Subra) Subrahmanyam, Jean Tirole, Dimitri Vayanos, and Glen Weyl. The advice of the editor and several anonymous referees
was very useful. We are thankful for the research assistance of Sergey Lobanov and Felipe Veras.
Strulovici acknowledges financial support from the National Science Foundation Grant 1151410.

© 2012 The Econometric Society

DOI: 10.3982/ECTA8822

2470

D. DUFFIE AND B. STRULOVICIlevel. We study the impact on capital mobility of search costs, discounting, asset
volatility, and other parameters.
This model is motivated by empirical evidence, some of which is reviewed
in the last section. The evidence is that supply or demand shocks in asset markets, in addition to causing an immediate price response, also lead to adjustments over time in the distribution of capital across markets and adjustments
over time in relative conditional mean asset returns. This asset price behavior
reflects delays in the adjustments of investors’ portfolios. We are particularlyinterested in how those adjustments are affected by the behavior of intermediaries.
An example is the limited mobility of capital into reinsurance markets, documented by Froot and O’Connell (1999), who wrote: “Our results suggest that
capital market imperfections are more important than shifts in actuarial valuation for understanding catastrophe reinsurance pricing. Supply, rather thandemand, shifts seem to explain most features of the market in the aftermathof a loss.” In subsequent work, Froot (2001) continued: “We    find the mostcompelling (evidence) to be supply restrictions associated with capital market
imperfections and market power exerted by traditional reinsurers.”
A significant body of theory examines the implications of search frictions for
asset pricing. For example, differences in search frictions across different asset
markets were treated by Weill (2008) and Vayanos and Wang (2007). Duffie,Gârleanu, and Pedersen (2005) studied the implications of search frictions ina single asset market with market making. In the context of a single asset market, Duffie, Gârleanu, and Pedersen (2007) and Lagos, Rocheteau, and Weill(2011) modeled recoveries in mean returns after a shock to the preferencesof investors, corresponding to a gradual reallocation of the asset to more suitable investors, rather than by cross-market capital dynamics as here. Earlier
search-based models of intermediation include those of Rubinstein and Wolinsky (1987), Bhattacharya and Hagerty (1987), Moresi (1991), Gehrig (1993),
and Yavaş (1996).
Related work on the implications of capital market frictions for asset pricing dynamics includes the models of Basak and Croitoru (2000) and He andKrishnamurthy (2012). In terms of some objectives and model features, independent work by Gromb and Vayanos (2012) is closely related to ours. Common to our models, local hedgers are immobile, while arbitrageurs can work
across markets, driving returns toward fundamental levels, subject to frictions
that prevent them from perfectly equating returns in the two markets. Our respective approaches, however, are quite different. We focus on the dynamics
of intermediation, capital movements, and risk premia.
We offer a quick synopsis of the model and results. In each of two markets
that are partially segmented from each other, a continuum of risk-neutral investors earn a mean rate of return that is strictly decreasing in the total amountof capital invested in that market. Segmentation of the two markets arises
from the limited capacity of intermediaries to contact individual investors in

CAPITAL MOBILITY AND ASSET PRICING

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the “over-capitalized” market and arrange for the movement of their capitalto the “under-capitalized” market, where they can currently earn higher returns. Intermediaries charge investors a fee for this service. The fee is a fraction of the gain in present value of future net payoffs to the investor that isachieved by the reallocation of capital. The investor and intermediary bear inmind that, due to the lack of perfect correlation in the future investment returns in the two markets, the market that is currently relatively over-capitalized
may later become relatively under-capitalized, at which point the investor will
wish to reallocate his or her capital to the original market, and so on. Intermediation costs are increasing in the effort made to contact investors. For each
investor, the conditional mean arrival intensity of contacts with the intermediary is strictly increasing in the effort of the intermediary. Intermediaries are
assumed to be unable to commit to their future intermediation effort policy.
In equilibrium, naturally, intermediation efforts are increasing in the degree
to which capital is imbalanced across the two markets. Thus, the differencein the conditional mean returns of the two markets is mean-reverting. In thesimplest parametric examples that we consider, intermediation efforts are minimal whenever the ratio of capital in the over-capitalized market to capital in
the under-capitalized market is below a threshold that is characterized analytically, and is maximal otherwise.
We are particularly interested in the impact of competition among intermediaries on the equilibrium degree of capital mobility, through two channels.
First, an intermediary does not internalize the entire impact of its search effort
on leveling the distribution of capital across markets because each intermediary gets only a fraction of the aggregate intermediation fees. This prompts
intermediaries to search more as the number of intermediaries increases, allelse equal. Competition among intermediaries has a second and potentially
offsetting effect on capital mobility through the impact of fee bargaining on
incentives to intermediate. We outline a model in which, as the number of intermediaries increases, but aggregate search capacity is kept constant, the bargaining power of each intermediary is reduced. We show that the second effectof competition, through reduced bargaining power, can dominate, so that insome cases increasing the number of intermediaries reduces capital mobility.
Section 2 describes the market setting. Sections 3 and 4 analyze the monopolistic and oligopolistic intermediation cases, respectively. Section 5 summarizes
the implications of our model for asset-price dynamics and provides some evidence regarding the premise or implications of our model. Proofs and several
extensions of the model are found in the appendices. Appendices D through L
are located in Duffie and Strulovici (2012), the Supplemental Material to thispaper.
2. THE MARKET SETTING
This section presents a stylized model for the endogenous adjustment of capital and risk premia across markets. There are three types of agents: (i) local

2472

D. DUFFIE AND B. STRULOVICI

hedgers; (ii) investors who provide risk-bearing to hedgers in each of two local markets; and (iii) intermediaries (such as asset managers) who provide thefee-based service to investors of moving capital from one market to another.
We fix a probability space (Ω F  P) and a common information filtration{Ft : t ∈ [0 ∞)} satisfying the usual conditions.2
In each of two financial markets, labeled a and b, a continuum (a non-atomic
measure space) of local risk-averse agents own short-lived risky assets. These“hedgers” are not mobile across markets. They can be viewed in this respect
as relatively unsophisticated in the use of cross-capital-market transactions,or as having high transactions costs for trading outside of their local markets.
A continuum of investors that supply capital have access to cross-market trading, subject to intermediation frictions to be described. These suppliers of capital are risk-neutral, and are therefore willing to invest in the risky assets that
hedgers sell, provided the risk premium is strictly positive. In an insurance context, one might think of these suppliers of capital as stylized versions of the“Names” that supply risk-bearing capacity to the insurance market known as“Lloyd’s of London.”
The total levels of capital available in the two markets at time t are Xat and
Xbt , respectively. Capital can be reinvested continually in time, at the discretion of each investor, in the short-lived assets that are continually made available for sale by hedgers. Each unit of capital that is currently invested in market
i at time t is paid cash at the equilibrium dividend rate π(Xit ), where π(·) is astrictly decreasing continuous function. The dividend rate π(Xit ) is continually
reset in double auctions at which the supply and demand for the asset in market i are matched at each point in time. In Appendix D, we provide an example
in which π(x) is the equilibrium insurance premium in a market with x units
of insurance capital.
In return for the dividend rate π(Xit ), the provider of each unit of capital inmarket i agrees to absorb proportional losses of capital at loss events that arrive according to a Poisson process. The proportional losses of different events
are independently and identically distributed. That is, the cumulative proportional loss in market i is a compound Poisson process ρi . (We later extend someof our basic results to allow ρi to be a general Lévy process, allowing, for example, some role for the diffusive action of Brownian motions.) One unit ofcapital invested at time t therefore pays the supplier of capital 1 + π(Xit ) dt attime t + dt, in the usual “instantaneous” sense, if there is no loss event, and if
there is a loss event, has a recovery value of 1 + ρit , where ρit is the jumpsize. The jumps of ρi are bounded below by −1, preserving limited liability. The
loss events have a mean arrival rate η and a loss-size distribution ν with meanν̄, so the mean loss rate is ην̄. The risk-neutral investors therefore optimally
supply all of their local capital inelastically so long as the mean rate of return2
See, for example, Protter (2004) for the usual conditions and for other standard properties ofstochastic processes to which we refer.

CAPITAL MOBILITY AND ASSET PRICING

2473

π(x) − ην̄ is strictly larger than their time preference rate r. This necessary
condition on an equilibrium cash payout function π(·) is satisfied in the casesthat we examine, as indicated in Appendix A.
Cash payouts to investors are not reinvested into the capital pool. This assumption simplifies the model, as it implies homogeneous capital dynamics.
This treatment of cash payouts is typical of asset-management contracts usedby private-equity partnerships.
We assume that ρa = a + c and ρb = b + c , where the market-specificcompound-Poisson processes a and b are independent and have the same
distribution. The independent component c is common to the two markets.
This symmetry simplifies the calculation of an equilibrium and has the further
illustrative advantage that any differences in the conditional expected returns
in the two markets are due solely to differences in the capital levels of themarkets. We briefly discuss the asymmetric case in Section 5.
If there were no capital-market frictions, investors would instantly move capital between the markets so as to obtain the higher dividend rate, and in doing
so would equate the dividend rates π(Xat ) and π(Xbt ), and thereby equate Xat
and Xbt at all times. Indeed, given the symmetrically distributed returns of thetwo markets, investors would do so even if they were risk-averse, provided thatthey have no other hedging motives.
Frictions in the movement of capital may, however, lead to unequal levelsof capital in the two markets, and therefore unequal mean rates of return. If,
for example, Xat < Xbt  then the conditional excess mean rate of return of therisky asset in market a exceeds that in market b by π(Xat ) − π(Xbt ), despite
the identical idiosyncratic and systematic risks of the two assets. Whichevermarket has “too much capital” receives the lower risk premium.
An investor chooses how to deploy reinvested capital between the two markets, subject to the available trading technology. Letting Ct denote the net cumulative amount of capital moved by a particular investor from market a into
market b through time t, this investor’s capital levels, WatC in market a and WbtC
in market b, thus satisfyCdρat − dCt
dWatC = Wat−

andCdρbt + dCt 
dWbtC = Wbt−
C
= lims↑t WitC is the level of capital “just before” time t. Capital canwhere Wit−
be moved only when in contact with an intermediary, as will be explained.
A model for a proportional intermediation-fee process K will be determinedin equilibrium. An investor is infinitely lived, and thus has a utility of
 ∞


 C

−rt
C
E
e
Wat π(Xat ) + Wbt π(Xbt ) dt − Kt− d|C|t 
0

2474

D. DUFFIE AND B. STRULOVICI

where |C|t denotes the total variation of C up to time t. For simplicity, we haveassumed that transactions costs are paid directly by investors, and not deductedfrom the capital moved from market to market.
Because there is a continuum of investors, each takes as given the total capital processes Xa and Xb of the respective markets.
Intermediaries contact investors in order to profit from fees for moving theircapital from one market to another. The intermediation fee is a fraction q of
the gain in present value to an investor associated with redeploying the investor’s capital. For now, we take this bargaining power q as an exogenous
parameter. We endogenize q in Section 4 for the case of multiple intermediaries.
Conditional on the intensity process λ for contacts of investors by intermediaries, investors are contacted pairwise independently3 at the conditional mean
rate λt . An intermediary’s rate of cost for applying contact intensity λt is cλt ,for some technological cost coefficient c ≥ 0. For example, doubling the expected rate at which investors are contacted costs the intermediary twice as
much.4 The maximum feasible contact intensity of the market is some constant
λ̄ > 0
The utility of an intermediary is the expected present value of intermediation fees net of intermediation costs. The intermediary maximizes this utility
with respect to the choice of intensity process λ. In the next section, we formalize this maximization problem in the context of a monopolistic intermediary,
in a Markov equilibrium setting. Later, we show how the solution of the monopolistic case can be exploited to more easily address the case of oligopolistic
intermediaries.
In the final section, we explain that our results extend easily to handle therandom arrival and departure of capital suppliers, and to allow capital suppliers to use their own personal technologies for trading, in addition to obtaining
intermediation services.
3. THE MONOPOLISTIC CASE
We begin with the case of a monopolistic intermediary. We restrict attention
to the illustrative example of an insurance market in which each loss event
affects only one of the two markets and results in a total loss of capital (ν̄ = 1).
Appendices treat more general cases, including partial recovery, loss events
that can affect both markets simultaneously, as well as proportional losses andgains that are based on Brownian motion.
3
That is, conditional on the path {λt : t ≥ 0} of the intermediation intensity process, the timesof contacts of any distinct pair of investors, i and j, are the event times of independent Poissonprocesses Ni and Nj with the common time-varying intensity process λ.
4This can be viewed as a contact technology in which the intermediary adjusts a “broadcast”
intensity, for example adjusting the rate of purchase of advertisements or other forms of marketwide intermediation efforts. This differs from a model in which, for example, contacting twice as
many individuals at a given intensity costs twice as much.

CAPITAL MOBILITY AND ASSET PRICING

2475

3.1. Equilibrium
We will define and characterize equilibria in which the intermediation intensity is of the symmetric Markov form λt = Λ(Xt  Yt ) for some measurable
policy function Λ : R2+ → [0 λ̄], whereXt = max(Xat  Xbt )
Yt = min(Xat  Xbt )
In equilibrium, at any time, only investors in that market with greater capital agree to have any of their capital moved to the other market. Because aninvestor has linear preferences, it is optimal when contacted to move either nocapital or all capital to the other market.5
In a manner similar to that of Weill (2007), the exact law of large numbers
allows us to calculate the aggregate rate of movement of capital. We let Waj (t)and Wbj (t) denote the levels of capital in markets a and b, respectively, of investor j at time t. Letting m(·) denote the non-atomic measure over the spaceof investors, the total rate at which capital is moved from market a to market
b is almost surely6
λt 1{Xat >Xbt } Waj (t) dm(j) = λt 1{Xat >Xbt } Waj (t) dm(j)= λt 1{Xat >Xbt } Xat 
Likewise, the rate at which capital moves from market b to market a is
λt 1{Xbt >Xat } Xbt 
The continuation value of an investor per unit of capital in the market withexcess capital can be represented as G(Xt  Yt ), for some G : R2+ → [0 ∞), andlikewise for the present value H(Xt  Yt ) of each unit of capital in the undercapitalized market.
These values, defined from primitive stochastic processes in Appendix A,
include the effects of future movements of capital to a market that is undercapitalized and, once that market becomes over-capitalized, back to the othermarket, and so on. Assuming differentiability, which we will verify in equilibrium, Itô’s formula implies that G and H are characterized as solutions to the5
If he or she has any capital in the market with more total capital, then all of this investor’scapital will be moved, provided the proportional transaction-costs process K is not too large, andthis is the case in any equilibrium for our model, as we shall see once the model is completely
specified. Thus, although we allow that a given supplier of capital may initially have nonzero
capital in both markets, all of his or her invested capital will optimally be held in just one of thetwo markets at any time after the first time of contact with an intermediary.
6
Sun (2006) showed that the probability space, the agent space, and the measurable subsetsof their product can be constructed so as to satisfy the exact law of large numbers. With this, thettotal quantity of contacts of agents by intermediaries up to time t is Nit dm(i) = 0 λs ds almost
surely.

2476

D. DUFFIE AND B. STRULOVICI

coupled equations(1)

rG(x y) = π(x) − Gx (x y)xΛ(x y) + Gy (x y)xΛ(x y)+ (1 − q)Λ(x y) H(x y) − G(x y)− ηG(x y) + η G(x 0) − G(x y) (2)

rH(x y) = π(y) − Hx (x y)xΛ(x y) + Hy (x y)xΛ(x y)+ η G(y 0) − H(x y) − ηH(x y)

where subscripts denote partial derivatives. The first of these equations statesthat in the over-capitalized market the rate of loss in value due to time preference, rG(x y), is equal to the total expected rate of net profit to investmentin that market. That rate of expected profit includes, first of all, the dividend
payout rate π(x). The next two terms are the rates of change of G(Xt  Yt ) due
to intermediated flows of capital out of the over-capitalized market and into
the under-capitalized market, respectively. The following term is the expected
rate of gain (1 − q)Λ(x y)(H(x y) − G(x y)), net of intermediation fees, associated with the chance to switch to the higher-premium under-capitalized
market. The final two terms reflect the expected rate of impact of loss events.
As the basic version of the model assumes no recovery value at these events,
ηG(x y) is the expected loss in value due to the occurrence of these loss eventsin the investor’s market. The final term η(G(x 0) − G(x y)) is the expected
gain to the investor associated with a loss event in the other market. The second
equation, for the total rate of investment return while in the under-capitalized
market, is similarly explained.
Fixing an intermediation policy Λ, the gain in present value per unit of redeployed capital at time t, before the associated fee, isF Λ (Xt  Yt ) = H(Xt  Yt ) − G(Xt  Yt )
Thus, at bargaining power q, the intermediary’s fee per unit of redeployed
capital is qF Λ (Xt  Yt ).
If forced to accept the fees associated with a conjectured intermediation policy Γ , a monopolistic intermediary’s optimal value for beginning with capitallevels x and y in the respective markets is
 ∞



−rt
Γ
(3)
e Λ(Xt  Yt ) Xt qF (Xt  Yt ) − c dt V (x y) = sup EΛ0

If a policy Λ that solves this problem is the same as conjectured policy Γ , thenwe say that Λ is an equilibrium. Thus, in equilibrium, fees are based on consistent conjectures by investors of the monopolist’s future intermediation intensity process.

CAPITAL MOBILITY AND ASSET PRICING

2477

We assume, and later verify, that the intermediary’s equilibrium initial value
V (x y) is finite and differentiable. The associated Hamilton–Jacobi–Bellman(HJB) equation, for fees determined by a conjectured intermediation policy Γ ,is(4)

0 = sup −rV (x y) + U (V  x y  Γ ) ∈[0λ̄]where, by Itô’s formula,

U (V  x y  Γ ) = −Vx (x y)x + Vy (x y)x+ η V (y 0) + V (x 0) − 2V (x y)+  xqF Γ (x y) − c 
PROPOSITION 1—HJB Equation: Given an assumed intermediation policy
Γ , suppose that V̂ is a bounded differentiable function satisfying the HJB equation (4). Then V̂ is the value function V of the optimization problem (3). Any measurable policy (x y) → Λ(x y) which, for each (x y), attains the supremum (4)is an optimal policy.
The proof, given in Appendix E.2, is by a traditional martingale verificationargument.
From the HJB equation, it is optimal for a monopolistic intermediary to
search at maximal intensity for opportunities to move capital whenever it isstrictly beneficial to search at all, and this is precisely when
(5)



c
qF Λ (Xt  Yt ) − Vx (Xt  Yt ) − Vy (Xt  Yt ) >Xt

The left-hand side is the fee per unit of capital moved, net of the associatedmarginal loss in present value of future intermediation fees. This loss in future
fees is caused by the associated reduction in heterogeneity of capital levels
across the two markets, which always lowers the gain to investors of shifting
their capital. The right-hand side of (5) is the cost of search per unit of capitalthat can be attracted.
To obtain the simplification associated with homogeneity,7 we suppose thatthe inverse demand function π(·) is of the form k0 + k/x for positive constants
k0 and k. As explained in the insurance setting of Appendix D, this can be arranged by suitable assumptions on the cross-sectional distribution of hedgers’
aversions to loss risk. Capital is therefore optimally reinvested locally, absent
an opportunity to move capital to a different market, so long as the mean rate
7
In Appendix A, some results are shown for the more general case π(x) = k0 + kx−γ for whereγ is any positive constant.

2478

D. DUFFIE AND B. STRULOVICIof return to investment exceeds the discount rate. Because investors will only
participate in local investment at a nonnegative risk premium, we haveπ(x) − η − r = k0 − η − r −

k
≥ 0
x

Because x is unbounded, it must therefore be the case that k0 ≥ r + η. Because
the constant k0 is common to the two markets, it has no effect on benefits
to investors of moving their capital. Our analysis of the equilibrium therefore
proceeds without loss of generality by ignoring k0 (treating it as though equal
to zero) until further notice.
Because the intermediary has linear time-additive preferences and becauseof the homogeneity of π, it is natural to look for equilibria for which the ratio
Zt = Xt /Yt of total capital in the over-capitalized market to total capital in theunder-capitalized market determines the optimal intermediation intensity, andwe focus on such equilibria from now on. A loss event causes the capital ratio
Xt /Yt to jump to +∞. While we allow this formally, the analysis can be done
similarly in terms of the ratio Yt /Xt , which remains in [0 1] almost surely, andour results apply with only notational changes.8
The homogeneity of the payout-rate function π and policy Λ implies that
H and G are homogeneous of degree −1. As a result, G(z 0) = g0 kz −1 forsome positive constant g0 to be determined. Letting f (z) = F Λ (z 1)/k andL(z) = Λ(z 1), homogeneity of F Λ and direct calculation from (1) and (2)imply that f solves the ordinary differential equation
0 = −rf (z) + 1 − z −1 − zL(z)f  (z) + −f (z) − zf  (z) L(z)z
(6)
 


− (1 − q)f (z)L(z) + η g0 1 − z −1 − 2f (z) The relevant boundary condition is f (1) = 0, corresponding to no gain fromswitching when the two markets have the same capital levels. The dynamic
equation for the homogenized gain from switching can be broken down as follows. The first term is a standard discounting term. The second term is thepremium differential across markets. The third and fourth terms capture the8Provided the initial conditions include a strictly positive amount of capital in at least one market, the probability that Xt and Yt ever reach zero at the same time is 0. The partial-recovery
case that we analyze in Appendix H has strictly positive capital levels in both markets at all times
after time zero, given a strictly positive level of capital in at least one of the markets at time zero.
In that context, the assumption that loss events strike only one of the two markets at a time iswithout loss of generality because any common jump component would have no effect on theratio of X to Y . The sole exception is a case of common jumps with a jump-size distribution that
supports −1, in which case there is a nonzero probability that Xt and Yt can be zero simultaneously. For the same reason, it is without loss of generality when characterizing equilibria that thereturn processes ρa and ρb have no drift, that is, a component that is linear in t. Any common
Brownian component is likewise irrelevant to optimal intermediation behavior.

CAPITAL MOBILITY AND ASSET PRICING

2479

impact of the switching activity on market heterogeneity. Because we consider
the homogenized version of the equation, this impact is twofold (any capitalincrease in the small market must be renormalized to 1). The fifth term captures investors’ gain from the switching activity, while the last term takes into
account capital losses on both markets. Our next result is that the switching
gain f (z) is strictly positive whenever capital levels are unequal.
PROPOSITION 2: Given any intermediation policy Λ, f (z) is strictly positive forz > 1. That is, given any Λ, investors in the over-capitalized market optimally accept the offer to move all of their capital out of the over-capitalized market whenever given the opportunity.
Taking F Λ as given, the optimal present value V of intermediation profits ishomogeneous of degree 0, that is, V (x y) = V (x/y 1) for y > 0. In particular,
the policy Λ achieving the supremum of the HJB equation (4) must also be homogeneous of degree 0; that is, Λ(x y) = L(x/y) for some L(·). Because theswitching-gain function f depends on the policy function L, the determinationof equilibrium is reduced to a fixed-point problem: Find a pair (f L) such that:(i) given f , the policy L is optimal, and (ii) given L, the marginal gain function
f is that determined by L through (6).
In Appendix A.3 (Proposition 12), we show that any equilibrium must be of
the “bang-bang” form:
Λ(x y) = 0

x < Ty= λ̄

x ≥ Tyfor some trigger ratio T ≥ 1 of the capital level in the over-capitalized market
to the capital level in the under-capitalized market. This is intuitive. Because
the HJB equation is linear with respect to the intensity chosen by the intermediary, we anticipate the optimality of switching from minimal to maximal
intensity whenever there is sufficient marginal gain from moving capital fromone market to the other. This occurs when the levels of capital in the two markets are sufficiently different. Our problem is reduced to finding an optimal
trigger ratio T , which then completely determines equilibrium behavior.9
The differential equation (6) for f now reduces to



(7)
r + 2η + λ̄ (1 − q) + z f (z) + λ̄(1 + z)zf  (z)1= (1 + ηg0 ) 1 − z ≥ Tz
9
If the intermediary could fully commit to a policy, it might be optimal for him to reduce
mobility strictly below full capacity.

2480
and
(8)

D. DUFFIE AND B. STRULOVICI1(r + 2η)f (z) = (1 + ηg0 ) 1 −z
z ∈ [1 T ]For z ∈ [1 T ], the solution is trivial:
1
1 + ηg0(9)
1−
f (z) =
r + 2η
z
In particular, we verify that f (1) = 0, consistent with the observation that thenet present value of moving capital from one market to the other is 0 when thelevels of capital in the two markets are the same.
We can rewrite (7) as


1
(10)
b z ≥ T
(a + z)f (z) + z(1 + z)f  (z) = 1 −zwhere a = (r + 2η + (1 − q)λ̄)/λ̄ and b = (1 + ηg0 )/λ̄.
Letting v(z) = V (z 1)/k, the HJB equation reduces to(11)

0 = sup −rv(z) − zv (z) − z 2 v (z)∈[0λ̄]





c
 
+ 2η v0 − v(z) + qzf (z) −kwhere v0 = V (y 0)/k = V (x 0)/k. Therefore,(12)v(z) = v1 z ∈ [1 T ]

where
(13)

v1 =

2η
v0 < v0 
r + 2η

and(14)

κv(z) + v (z)z(1 + z) = d + qzf (z)z ≥ T

where κ = (r + 2η)/λ̄ and
d=

c
2ηv0
− 
k
λ̄

Appendix A.2 contains a proof of the following monotonicity and regularity of v(·). Monotonicity of the value v(z) in the capital heterogeneity measure z is not an obvious result, in particular because the switching gain f (z) isCAPITAL MOBILITY AND ASSET PRICING

2481

not, in general, monotonic. That is, fixing the capital level y = 1 in the market
with less capital, the marginal gain f (x) from switching capital from the overcapitalized market to the under-capitalized market need not be monotone in xeven though the increase in payout rate π(y) − π(x) is strictly monotone in x.
This is the case because, as x gets large, both g(x) and h(x) ≡ H(x 1)/k go
to zero, and so must therefore f (x) = h(x) − g(x) go to zero. Intuitively, as
x gets large, the global amount of capital is large, and although it can be intermediated from one market to another, the value of being a capitalist is not
attractive when there is “too much” capital relative to the demand by hedgers
to lay off risk.
The intuition for monotonicity of v(z), however, is that, for any assumed
trigger ratio T , optimal or not, the total rate of activated intermediation feesqλ̄zf (z), per unit of capital in the over-capitalized market, is not relevant on[0 T ] by definition of T , and is strictly increasing in z above T . In particular, even though f (z) need not be monotone in z, zf (z) is monotone in z, as
shown in Appendix A.2. Intuitively, zf (z) is the aggregate trade surplus generated by the intermediary’s activity, and that surplus is increasing in market
heterogeneity.
PROPOSITION 3—Value Function Monotonicity: For any trigger capital ratio
T , the solution v of (11)–(14) is bounded, increasing, and strictly increasing on[T ∞).
The smooth-pasting condition v (T ) = 0 is equivalent to(15)

qTf (T ) =ckand implies the trigger capital ratio
(16)

T =1+

c(r + 2η)
(1 + ηg0 )qk

To identify the constant g0 , we use a conservation equation: The sum of thevalue functions of all investors and of the intermediary must be equal to thepresent value of all cash dividend payments net of the search costs incurred by
the intermediary. After calculations shown in Appendix A.1, this conservation
principle is equivalent to
(17)

kg0 =


2 c λ̄ −
1 − e−(2η+r)a(T ) − kv0 rr

where a(T ) = log(1 + 1/T )/λ̄.
A proof of the following result guaranteeing the existence and uniqueness ofa trigger strategy is found in Appendices A.4 (existence) and A.5 (uniqueness).

2482

D. DUFFIE AND B. STRULOVICIPROPOSITION 4—Existence and Uniqueness: There exists a unique trigger
capitalization ratio T satisfying (17), (7), (8), and (16).
An iterative algorithm for computing the equilibrium is presented in Appendix F, whose steps exploit the previous equations as follows: starting fromsome initial level of v0 , Equations (16) and (17) are used to determine g0 andthe threshold T . Using these values, the ordinary differential equation (14) isused to determine a new value of v0 as the limit of v(z) as z goes to infinity,
concluding the first iteration of the algorithm.
This analysis leads to the following characterization of equilibrium, which
includes the result that, in the absence of search costs, the intermediary does
not exploit his position to restrict movement of capital, but rather provides
maximal intermediation, nevertheless generating fee income from his or her
imperfect ability to instantaneously move capital from one market to the otherdue to the upper bound λ̄ on contact intensity. As λ̄ becomes large, the capitallevels will be nearly equated across the two markets at all times, as in a completely frictionless market, and in the limit there would be no intermediation
rents.
PROPOSITION 5: Suppose that the payout-rate function π is of the form π(x) =k0 + k/x. Then there exists a unique equilibrium. In equilibrium, there is no intermediation (λt = 0) whenever the ratio of capital levels in the two markets is
between 1/T and T , for a uniquely determined capital-ratio trigger T . Otherwise,intermediation is at full capacity (λt = λ̄). The trigger ratio T is given by (16),where the constant g0 is given by (17). If there is no intermediation cost (c = 0),then the intermediary always works at full capacity (i.e., T = 1).
3.2. How Intermediation Depends on Market Parameters
We turn to comparative statics, focusing on the behavior of the thresholdcapital ratio T . A higher trigger ratio T corresponds to less intermediation,
because the intermediary waits until Xt /Yt exceeds T before becoming maximally active. We therefore define capital mobility to be increasing in a parameter whenever the equilibrium threshold T is decreasing in that parameter. We
show that these comparative statics carry over to the oligopolistic case.
PROPOSITION 6—Comparative Statics for c k η and r: Capital mobility isdecreasing in the intermediation cost coefficient c, increasing in the payout-rate
coefficient k, and decreasing in the discount rate r. Fixing the dividend payout
rate function π(·), capital mobility is decreasing in the loss-event intensity η.
A proof is provided in Appendix G. It is intuitive that increasing the costs ofintermediation, represented by c, reduces the amount of intermediation provided in equilibrium. Once the trigger T is chosen, the costs of search intensity

CAPITAL MOBILITY AND ASSET PRICING

2483

are borne entirely by the intermediary. Other things equal, raising c therefore
lowers the desire to search. The coefficients c and k affect the trigger T only
through the ratio c/k, explaining the comparative static for k. Intuitively, multiplying c and k by the same factor amounts to a change of currency unit, which
is why only the ratio c/k affects the equilibrium threshold. The comparative
statics for η and r are more subtle because, for a given trigger ratio T , both ηand r have an effect on the value functions of investors. Equation (16) is the
key to understand these comparative statics. A higher after-shock continuation
value for investors (i.e., a higher g0 ) results in a lower T and thus a higher capital mobility. A higher r has two effects on investors. The direct effect is thatit reduces their gain from switching capital, which is the role of r in the numerator of (16). The indirect effect is that, because the present value of gainfrom switching capital is lower for higher r, investors pay lower fees to the intermediary, resulting in a higher g0 . As we show in our proof, the former effectdominates the latter, so T is decreasing in r. One intuition is that the second
effect is an indirect consequence of the first, direct effect, so we may expect the
net comparative static to follow the first effect.10
Our proof likewise shows that T is increasing in η, holding π fixed. Of
course, increasing the mean loss frequency η would naturally raise the equilibrium loss insurance rate π(Xit ), which on its own would tend to increase intermediation (lower T ). To analyze this effect, we write π(x) = k0 (η) + k(η)/xto show the dependence of the coefficients k0 (η) and k(η) on the mean loss
rate η. The coefficient k0 (η) plays no role in intermediation gains. The impact
on intermediation intensity of replacing η with some η > η is thus equivalent
to the effect of leaving π unchanged and replacing the cost coefficient c withc  ≤ ck(η)/k(η ). This effect can be small or large, depending on the sensitivity of k(η) to η, reflecting how the elasticity of hedging demand varies with the
expected loss frequency. Thus, the overall impact on intermediation intensityof a given increase in the loss-event intensity η is to lower intermediation incentives (raise T ) precisely when the impact on k(η) is sufficiently small, andotherwise to raise intermediation intensity.
Increasing the bargaining power q of the intermediary increases the fractionof the gains to trade that goes to the intermediary, prompting the intermediaryto search for more investor capital to move, thus setting a lower trigger ratio T ,holding constant the gains to investors for moving capital. Obviously, however,raising q lowers the present value of investors associated with future movements of capital, thus lowering the amount of gain they have to share with theintermediary. The proof in Appendix G of the following result demonstrates
that the direct effect dominates the indirect effect on the investors’ values, provided that q remains below 1/2.
10
The techniques used for the comparative statics of discounting draw on those introduced by
Quah and Strulovici (2009, 2012)

2484

D. DUFFIE AND B. STRULOVICIPROPOSITION 7—Comparative Statics for q: Capital mobility is increasing in
the bargaining power q of the intermediary, for 0 < q < 05.
The impact on capital mobility of the capacity λ̄ for search intensity depends
on other parameters, and particularly on the discount rate. There are situations in which lowering λ̄ lowers the trigger ratio T , leading the intermediary
to work more often, albeit with a lower capacity. We provide examples andsome intuition for the fact that, depending on the discount rate r, the threshold capital ratio T can either increase or decrease with capacity. We focus on
the impact of λ̄ on the threshold T , noting that the level of λ̄ has also a direct
impact on capital mobility, through the rate at which capital moves from one
market to another, when it does.
We argue by continuity from the case of η 0. Taking π(x) = k0 (η) +k(η)/x, we can create a family of economies that keep k(η) fixed at somek̄ as η varies near zero by increasing the loss aversion of hedgers. With this, asη approaches zero, the equilibrium intermediation policy converges to a policythat moves capital until the threshold T is reached (with no changes in capitallevels afterward), so the associated trigger ratio converges to
(18)

T =1+

cr
k̄q



which is independent of the capacity λ̄. Moreover, the associated gain functions
converge to a gain function f with11f (T ) =1−rT
Suppose first that agents are patient (have a low discount rate r). As λ̄ goesup, capital heterogeneity goes down, hence the value of moving capital is lower.
Moreover, a higher-capacity intermediary will more quickly run out of capital
to be moved, and thus stop receiving fees earlier. Overall, this implies by continuity that we can construct example economies with small η such that, as λ̄
is reduced and other parameters are held constant, the intermediary receives
more fees, and for longer, and hence the value function of investors is lower,
resulting in a higher threshold.11
Now, we consider the opposite case of nearly myopic investors (high r). The
previous argument breaks down: Investors do not care much about future heterogeneity; they care mainly about the immediate gain from switching, which
depends on current heterogeneity. Moreover, the immediate fees are increasing with capacity λ̄. A higher-capacity intermediary therefore receives higher
11
See Equation (16), which shows that T is decreasing in investors’ after-shock continuation
value g0 .

CAPITAL MOBILITY AND ASSET PRICING

2485

immediate fees, which reduces investors’ overall value of switching capital andincreases the threshold.
Numerical examples support this intuition. For q = 05, c = k̄, and r = 004,
for η sufficiently small and k(η) = k̄ sufficiently insensitive to η, the capitaltrigger T decreases with λ̄ for 001 < λ̄ < 05. For the same q and c but taking
r = 10, however, the trigger ratio T increases with the intermediation capacityλ̄, over the same interval.
4. INTERMEDIARY COMPETITION
We now solve for equilibria with oligopolistic or perfectly competitive markets for intermediation.
There are two channels through which intermediary competition affects theequilibrium level of intermediation offered by the market. First, an intermediary internalizes the impact of intermediation intensity on the heterogeneityof capital levels across the two markets, and thus the impact on the intermediary’s future profits. A given intermediary does not, however, internalize theeffect of increasing its intermediation on reducing the profit opportunities ofother intermediaries. Through this first channel, increasing the number of intermediaries should therefore weakly increase the total amount of intermediation. In the benchmark case in which a loss event destroys all capital in theaffected market, there is nothing to internalize, because the after-shock situation is independent of the pre-shock heterogeneity level, and our upcoming
Proposition 8 states that the threshold is independent of the number of intermediaries. In Appendix J, we explain why there is a strict increase in capital
mobility from increasing competition if loss events leave some residual capital, because in this case the post-loss heterogeneity depends on the pre-loss
heterogeneity.
The second channel by which increasing competition changes capital mobility is as follows. When in contact with an investor, an intermediary considers
the ability of the investor to compare the intermediation fee offered with thefees offered by other intermediaries. This plays a role, in an extension of themodel that is offered later in this section, in determining the effective bargaining power of the intermediary, and through that channel, has an impact on the
profitability of intermediation. We start by taking bargaining power as fixed
and then endogenize the fees received by the intermediary. As we shall see,in some cases the second channel can dominate the first, that is, in which anincrease in the number of intermediaries lowers capital mobility.
4.1. Intermediary Competition at Fixed Bargaining PowerFor a given bargaining power q, we show that equilibrium trigger policies
for the oligopolistic case can be translated directly from the case of monopolistic intermediation through a simple inspection of the associated Hamilton–

2486

D. DUFFIE AND B. STRULOVICIJacobi–Bellman equations. From this, we also obtain comparative statics directly from those of the monopolistic case.
For the oligopolistic case, we take n identical intermediaries, each with an
upper bound λ̄/n on intermediation intensity, and with the same proportional
cost c of intermediation. The monopolistic case (n = 1) is the special case considered in the previous section. Thus, all cases have the same feasible market
dynamics and costs.
We consider only Markov time-homogeneous equilibria. Equilibrium incorporates the degree to which intermediaries internalize the impact of their intermediation intensity on the heterogeneity of capital levels across markets.
For an oligopolistic equilibrium in trigger strategies, each of the n intermediaries has a reduced value function v, with v(z) = V (z 1)/k, that solves the
reduced HJB equation, extending the monopolistic version (11):(n − 1)λ̄1{z≥T } −  zv (z)(19)
0 = sup −rv(z) + −n∈[0λ̄/n](n − 1)λ̄z1{z≥T } + z zv (z)
−
n




c
 
+ 2η v0 − v(z) + qzf (z) −kreflecting the presumption by the given intermediary that the n − 1 other intermediaries have adopted a specific trigger capital ratio T . The equilibrium
condition is that the same trigger policy is optimal for the given intermediary.
Verification of the HJB solution as the value function and of the optimality of
the associated candidate policy function is as for the monopolistic case.
Thus, an equilibrium for the n-intermediary problem is again given by bangbang control for all intermediaries, each exerting no effort when Zt < T andmaximal intermediation intensity λ̄/n whenever Zt ≥ T , for a trigger capitalratio T . We show that optimality implies that there is no intermediation at or
below the capital ratio T satisfying the smooth-pasting condition v (T ) = 0.
This, along with (19), implies that(20)

qTf (T ) −

c
= 0
k

From (20), we see that an intermediary’s optimization problem in a setting with n intermediaries is equivalent to that of a monopolistic intermediary with maximum intermediation intensity λ̄. Indeed, for a given threshold
T , the monopolistic and oligopolistic cases yield the same function f determining proportional intermediation fees, and hence the same smooth-pasting
condition (20).
In fact, this is actually the unique equilibrium, even allowing for the possibility of non-trigger strategies! To see this, consider any Markov equilibrium,CAPITAL MOBILITY AND ASSET PRICING

2487

not necessarily of the trigger-ratio form, and let f denote the function determining the associated gain from switching. An intermediary’s HJB equation
is of the form (19), except that (i) the aggregate of other intermediaries’ contact intensities may be almost arbitrary, and (ii) the value functions may varyacross intermediaries. Owing, however, to the form of the HJB equation, theindifference condition is nevertheless given by (20), and thus is the same forall intermediaries. This shows that any Markov equilibrium must be symmetricand of the trigger form.12 In fact, repeating arguments from the monopolistic
case leads to the following proposition.
PROPOSITION 8: With n intermediaries, there is a unique Markov, homogeneous equilibrium. This equilibrium is symmetric and determined by a trigger capital ratio equal to that of a monopolistic intermediary with the oligopolistic maximal contact intensity λ̄.
When shocks lead to a partial recovery of capital, however, the intermediation trigger capital ratio T is strictly decreasing in the number n of intermediaries, as shown in Appendix J.
4.2. Endogenous Bargaining Power
Competition to supply intermediation may also have an impact on an intermediary’s share of gains from trade when in contact with an investor. We next
consider the implications of market structure for the intermediary bargaining
power q. With n > 1 intermediaries, we suppose that some fraction ψn of investors are “well connected,” meaning that as they prepare to move capitalfrom one market to another, they are in simultaneous contact with more than
one intermediary. The number of intermediaries with whom a given investor isin contact could also be random, exploiting the law of large numbers, in whichcase ψn can be taken to be the probability that an investor, when contacted,
is in contact with more than one intermediary. Intuitively, a well-connected
investor has more bargaining power than a “captive” investor, one who is incontact with only one intermediary.
When modeling this intuition with a bargaining game, an issue is whether thecontacted intermediary is assumed to know whether the investor is in contact
with other intermediaries. We take this case.13 Another modeling approach12

The trigger form comes from showing, as in the monopoly case (Lemma 2), that the functionz → zf (z) is increasing.
13
It would be possible to allow for one-sided information. The fees derived could be obtained
as equilibrium outcomes of a bargaining process, although there may be additional equilibria.
See, for example, Sutton (1986). For an alternative approach to treating uncertainty about thedegree to which an intermediary’s customer is in contact with other intermediaries, see Green(2007).

2488

D. DUFFIE AND B. STRULOVICI

is a multilateral bargaining game with complete information, as in Stole andZwiebel (1996).
We consider a bargaining procedure à la Rubinstein (1982), in which theinvestor and a particular intermediary alternate offers. In our continuous-time
setting, the times between offer rounds can be treated as arbitrarily small, so
the inter-round discount factor can be taken to be 1. In that case, the investor
and intermediary agree immediately to split the surplus according to the Nashbargaining solution. The investor’s share depends on his outside option. If theinvestor is captive, his outside option is simply G(x y), the value of remaining
in the over-capitalized market. Thus, the normalized Nash product associatedwith a proportional fee of s isv(z) + s − v(z) h(z) − s − g(z) which is maximal at s = f (z)/2, corresponding to q = 1/2, meaning an equal
splitting of the gains with the intermediary.
For a well-connected investor, the normalized Nash product is
v(z) + s − v(z) h(z) − s − g(z) − (1 − q0 )f (z) 
where q0 is the equilibrium proportion of the gain from trade that the investor
would pay to another intermediary if this first round of bargaining were to
break down. The Nash product is maximized at s = q0 f/2, yielding an equilibrium intermediary share of q = 0 corresponding to the extraction of all surplus
by the well-connected investor.14
If the number of intermediaries in contact with the investor is known only by
the investor, then q is similarly obtained, and depends on the probability thatthe investor is captive.
Assuming pairwise independence of the connectedness of individual investors, the average of an intermediary’s share of gains across the infinite population of investors is almost surely(21)

11 − ψnq(n) = 0 × ψn + (1 − ψn ) =22

In particular, q(n) is decreasing in n if ψn is increasing in n. Obviously, ψ2 ≥
ψ1 . Going beyond the case of n = 2, it is somewhat intuitive that an investor
is more likely to be well connected as the number of intermediaries increases.
Appendix B briefly outlines a model with this natural feature.
Noting from (21) that q(n) < 1/2, Proposition 7 hints that lowering q(n) reduces an intermediary’s incentive to search, all else equal, because, for given
14
Another way to obtain this prediction is to assume that intermediaries connected to a given
intermediary post prices and engage in Bertrand competition. We are grateful to a referee forsuggesting the alternative of Bertrand competition.

CAPITAL MOBILITY AND ASSET PRICING

2489

capital dynamics, lowering q(n) reduces intermediation profits, and therefore
lowers the marginal benefit of raising intermediation intensity. We will next illustrate the second channel through which oligopolistic intermediation affects
capital mobility: By reducing each intermediary’s bargaining power, the incentive to intermediate is lowered.
Endogenous bargaining leads to complex dynamics, in which the number ofintermediaries actively searching for capital varies over time. To see this, consider a candidate equilibrium in which n intermediaries search at full capacity
whenever z > T , and no intermediary searches when z ≤ T . If a single intermediary deviates by searching for capital when z is in a left neighborhood of
T , then his fee per unit of capital switched is that of a monopolist, not that of
the n-intermediary case. This increases the value of this deviation. Despite thisadded complexity, we now show that oligopolistic intermediation may reduce
capital mobility.
4.3. Reduced Capital Mobility With More Intermediaries
A Markov strategy profile for n intermediaries consists of functions L1  L2     Ln on [1 ∞) into [0 λ̄/n]. Here, Li (z) denotes the search intensity ofintermediary i when the heterogeneity of capital across the two markets is z =x/y. The associated aggregate capital mobility is
L(z) =nLi (z)

i=1

To exploit the fee share q(n) derived above, we focus on simple strategies,
for which Li (z) is either 0 or λ̄/n. With this restriction,15 we can associatewith any strategy profile an increasing sequence T0  T1      TJ of capital-ratio
thresholds with the property that, whenever the capital ratio Zt is in [Tj  Tj+1 ),a particular set Nj of intermediaries is active. We let nj = |Nj | denote the number of intermediaries in Nj .
Using our previous analysis of the oligopolistic case with fixed bargaining
power, we say that a profile of simple strategies is a Markov equilibrium if, forall j and z ∈ [Tj  Tj+1 ),(22)

q(nj )zf L (z) −

c
≥ 0
k

and
(23)

q(nj + 1)zf L (z) −

c
≤ 0
k

15
Extending the analysis to general Markov strategies would be possible if one computes, forany possible strategy, the expected fee for each intermediary as a function of his search intensity
and of the aggregate search intensity.

2490

D. DUFFIE AND B. STRULOVICI

where f L (z) denotes the marginal gain to an investor from switching to themarket with less capital, given an aggregate intensity policy L.
The first inequality means that any intermediary in Nj contacting investors at
capital ratio z does so optimally, given equilibrium fee share q(nj ). The second
equation states that any intermediary in Nj + 1 is optimally not contacting at
capital ratio z, given the equilibrium fee share q(nj + 1) that he would get if hesearched.
We let T̄ = inf{z̄ : L(z) = λ̄ z ≥ z̄}, the smallest level of capital heterogeneity
above which intermediaries search at full capacity. We denote by T1 the monopolistic threshold. For the result to follow, recall that η is the mean arrival
rate of loss events and that T̄ depends, through L, on the particular Markov
equilibrium under consideration. In any equilibrium, zf L (z) is strictly increasing in z. (For this, see the proof of Proposition 10.) This fact and Equations(22) and (23) imply that when q(n) is decreasing in n (i.e., a more connected
investor pays a lower fee), the number of active intermediaries is increasing in
z in any equilibrium.
PROPOSITION 9—Monotonicity: Suppose that q(n) is decreasing in n. Then,nj is increasing in j.
If q(n) is strictly decreasing in n, it is possible to prove by a fixed-point argument the existence of a Markov equilibrium, in which, whenever the capitalheterogeneity Zt is [Tk  Tk+1 ), precisely the first k − 1 intermediaries are active,
and the remainder are inactive.16
The following result applies to all equilibria.
PROPOSITION 10: Suppose that the number n of intermediaries is at least 2.
There exists some η̄ > 0 such that, for any loss event intensity η ∈ (0 η̄) and any
associated Markov equilibrium trigger capital ratio T , we have T1 < T̄ .
This result states that, for sufficiently infrequent loss events, the reduced
bargaining power caused by oligopolistic competition reduces the domain of
16
For a sketch of the proof, let T denote a vector (T0      Tn ) in Rn+1 of capital heterogeneity
thresholds, with increasing elements. We let LT and f T denote the associated capital-mobility
and switching-gain functions. Taking T as given, a vector of best-response thresholds is uniquelydetermined by Equations (22) and (23). The best-response function of the respective intermediaries induces a new vector B(T) of capital heterogeneity thresholds. It can be checked that B(·) is
continuous. For this, we endow Rn+1 with the Euclidean topology and the space of switching-gain
functions (f L ) with the sup-norm topology. We have continuity of T → f T because of Equations(22) and (23), and because the best-response thresholds depend continuously on f L . Moreover,the best-responses preserve threshold monotonicity. Further, the best-response thresholds are
uniformly bounded below by T = rc/(kq(n)2) (using that f L (z) ≤ 2/r for all z), and bounded
above by T̄ = c/(kq(1)f (T )), ¯where f (z) is the gain from switching capital when capital moves¯¯
f (z) ≤ f L (z) for all L and z. The result follows fromat full capacity λ̄ until T¯ = 1, and satisfies¯
Brouwer’s theorem.

CAPITAL MOBILITY AND ASSET PRICING

2491

maximal capital mobility relative to that of the monopolistic case. Our assumption of a sufficiently small mean arrival rate η of loss events exploits the factthat intermediation fees are generated from the time of each loss event until
capital is sufficiently equalized across the markets. Although there are technical steps in the proof of this proposition, found in Appendix C, the argument
relies on a bound on improvements in the present value of intermediation fees
as one changes from one market setting to another. A simple way to provide
such a bound is thus to control the speed with which new fee-generating loss
events occur.
Proposition 10 shows that oligopolistic competition results in less intermediation than achieved by a monopolist, for some range of market heterogeneity.
This does not, however, rule out intermediation by oligopolists at capital ratios
below the monopolistic trigger level. The next result shows that, provided thatloss events are not expected too frequently, oligopolistic and monopolistic settings lead to a cessation of intermediation at approximately the same levels ofmarket heterogeneity.
For any n-intermediary Markov equilibrium with aggregate intermediation
strategy L, let
Sn = inf z : L(z) > 0 
the smallest heterogeneity level above which capital is mobile. A proof of thenext proposition may be found in Appendix C. We will rely on a sufficiently
small loss-event intensity for the same bounding effect explained after thestatement of Proposition 10.
PROPOSITION 11: For any ε > 0, there exists a strictly positive η̄ such that, forany mean loss-event rate η ∈ (0 η̄) and any associated Markov equilibrium withn players, we have Sn ≥ T1 − ε.
Propositions 10 and 11 together show that capital mobility is lower, at any
levels of capital, with oligopolistic intermediation than with monopolistic intermediation, provided that loss events are sufficiently infrequent.17
5. DISCUSSION AND CONCLUDING REMARKS
In a neoclassical model of asset markets, investors continually adjust theirportfolios so as to achieve the highest possible mean return for a given typeof risk. In equilibrium, an investor bearing a given type of risk is therefore
17
For the last two results, we have held constant the dependence of the dividend rate function
π(x) on the capital level x as the mean rate η of loss events is varied. We have the freedom,
however, of varying the population of hedgers as η changes, so as to offset the impact of variationof η on π(x), thereby satisfying the stated comparison between monopolistic and oligopolistic
intermediation for each fixed economy.

2492

D. DUFFIE AND B. STRULOVICIcompensated by a unique associated excess mean rate of return, no matterthe asset that carries the risk. In practice, however, investors make portfolio
adjustments with delays. In our model, the mobility of capital across different
investments is increased through the equilibrium efforts of intermediaries. Ourmodel has several implications for asset-price dynamics:
1. With unexpected changes in the amount of capital that is available tobear the risk represented by an asset, risk premia adjust more severely to capital shocks than in a neoclassical (perfect-mobility) setting, and then revert
somewhat over time as capital is redeployed. This time signature, present inessentially any setting with slow moving capital, is dampened to the extent thatintermediaries are active. Consider, for example, our simplest setting in which
capital levels change only at the times at which all capital in a given asset market is lost. Perfect capital mobility (c = 0 λ̄ = ∞) implies that Xat = Xbt for
all t. A loss event at time τ in market a would cause half of the capital frommarket b to move instantly to market a, so risk premia in both markets jumpup by
π(Xaτ ) − π(Xaτ− ) =

k

Xaτ−

and then remain constant until the next loss event. With imperfect mobility, however, the risk premium in market a would jump immediately to +∞and then decline at the rate λ̄Zt Xkat until the capital heterogeneity ratio Zt =Xbt /Xat drops to T or until another loss event occurs.
2. The degree to which risk premia vary across assets, after controlling forother determinants of risk premia, is increasing in the degree to which capitalis heterogeneously distributed across assets.
3. The speed of reversion of risk premia across assets toward common levels(after adjustment for other determinants of risk premia) is decreasing in thecost of intermediation. Increasing intermediation capacity, however, can either
increase or reduce capital mobility, depending on the setting, as explained in
Section 3.2.
4. All else equal, increasing the fraction of gross gains from moving capitalthat accrue to intermediaries increases the speed of adjustment of capital andrisk premia.
5. Lowering time discount rates increases the mobility of capital.
6. Increasing the volatility of asset returns, represented in our model by themean frequency η of loss events, can either increase or decrease the mobilityof capital through intermediation, depending on the relative magnitudes of theeffects of (i) raising neoclassical risk premia (thus increasing the rents available
to intermediaries), and (ii) increasing the volatility of capital heterogeneity at
a given level of intermediation, which lowers the incentive to intermediate.
7. Increasing the scope for intermediary competition by splitting the market
among more intermediaries can increase or decrease the equilibrium provisionCAPITAL MOBILITY AND ASSET PRICING

2493

of intermediation, depending on the relative magnitudes of the effects of (i) reducing the concern of an intermediary regarding the impact of its own activity
on lowering capital heterogeneity, and (ii) lowering the bargaining power of
an intermediary vis à vis its customers because of increased competition withother intermediaries. As the number of intermediaries increases, the former
effect raises intermediation incentives, while the latter effect can lower intermediation incentives.
Our introduction uses the example of the market for catastrophe risk reinsurance. Froot and O’Connell (1999), Zanjani (2002), and Born and Viscusi(2006) explained how premia for catastrophe risk insurance typically increase
dramatically when insurance and reinsurance firms suffer significant damage
claims after natural disasters, such as Hurricane Andrew in 1992. Then, overmany months, premia drop toward “soft-market” levels (absent other shocks tothe capital of insurers) because the replacement of insurance capital is delayed
by institutional barriers to capital raising, including the time spent searchingfor suitable new investors. According to Enz (2001), premia swing up and down
by as much as 50% over multiyear periods, and are closely linked with changes
in the capital levels of insurers, regardless of whether these changes in capitalare caused by damage claims or by unexpected returns to the asset portfolios
of insurers. From this, we know that the dynamics of insurance premia after amajor natural catastrophe are not caused mainly by inference concerning thearrival rate of future such events. We also know that most of the observed priceimpacts are not caused by inference about losses because major changes overtime in insurance premia following shocks to capital levels are highly correlated
across all major lines of property insurance covered by the same pools of capital covering catastrophe risk.18 These other lines cover, for example, aviation,
marine, motor, and proportional property. The link tying premium dynamics
across the various lines of insurance is the level of capital commonly available
to bear losses. Apparently, these various property-related insurance markets
are not segmented from each other by capital immobility. Rather, they are collectively segmented from other types of capital markets. Froot and O’Connell(1999) emphasized the slow speed of capital replacement as the major causeof slow premium adjustments.
That there is scope for intermediaries to mobilize dormant capital is apparent from a significant body of evidence that, when left on their own, many
individual investors adjust their portfolios remarkably infrequently. For example, Ameriks and Zeldes (2004) reported that, over a 10-year period, 44% ofinvestors in their sample made no changes to their portfolio allocations, andthat an additional 17% of these investors made a single reallocation during
this period. Mitchell, Mottola, Utkus, and Yamaguchi (2006) found that, of
1.2 million U.S. employees covered by over 1,500 401(k) investment plans, approximately 80% initiated no trades over a two-year period, while an additional
18

See, for example, Enz (2001, p. 5, Figure 1).

2494

D. DUFFIE AND B. STRULOVICI

10% made only a single trade.19 Based on our theoretical results, one presumes
that investor inattention is more evident in the data when the cost of contacting
individuals and deploying their capital is large relative to the potential associated intermediation profits. For example, high-net-worth individual investors
and institutional investors are likely to receive more attention from intermediaries because of the amounts of capital they can deploy and the associatedhigher total intermediation fees, than are most of the smaller investors whose
inattention is documented in this literature. This is consistent with evidence
provided by Bilias, Georgarakos, and Haliassos (2010) and Feldhütter (2009).
Further, variation across investors in financial sophistication (not captured in
our model) may lead to a negative correlation between the cost of achieving
a given level of contact intensity and the level of deployable capital. In ourmodel, intermediaries cannot differentiate among subclasses of investors.
Delays in processing information for purposes of investment decisions arealso in evidence from “price momentum” following fundamental news, as
documented empirically by Chan (2003), Engelberg, Reed, and Ringgenberg(2012), Dellavigna and Pollet (2009), and Cohen and Lou (2012), among others. Given that most financial intermediaries are themselves likely to receiveand process fundamental news quickly relative to the time for price reactions
documented in this literature, one presumes that there are limits on the average speed with which they can attract potential investors to such investment
opportunities. This inference is also consistent with numerous examples ofslow price adjustments to supply shocks in equity markets, including those ofHolthausen, Leftwich, and Mayers (1990), Scholes (1972), Coval and Stafford(2007), Andrade, Chang, and Seasholes (2008), Kulak (2008); with respect tosupply shocks caused by index recompositions, see Shleifer (1986), Harris andGurel (1986), Kaul, Mehrotra, and Morck (2000), Chen, Noronha, and Singhal(2004), and Greenwood (2005); and with respect to the expiration of commodity futures contracts, see Mou (2011).
In corporate bond markets, which are not traded on a central exchange, one
observes large price drops and delayed price recovery in connection with major
downgrades or defaults, as described by Hradsky and Long (1989) and Chen,
Lookman, and Schürhoff (2008), when certain classes of investors have an incentive or a contractual requirement to sell their holdings. Mitchell, Pedersen,
and Pulvino (2007) documented the effect on convertible bond hedge fundsof large capital redemptions in 2005. Convertible bond prices dropped andrebounded over several months. A similar drop-and-rebound pattern was observed in connection with the LTCM collapse in 1998. Newman and Rierson(2003) showed that large issuances of European Telecom bonds during 1999–
2002 temporarily raised credit spreads throughout the sector, evidence that it
takes time for intermediaries to locate long-term investors.
19
For further evidence on the slowness of individual portfolio adjustments, see Lusardi (1999,
2003), Brunnermeier and Nagel (2008), and Bilias, Georgarakos, and Haliassos (2010).

CAPITAL MOBILITY AND ASSET PRICING

2495

In these examples, the time pattern of returns or prices after a supply or demand shock reveals that the friction at work is not merely a transaction costfor trade. If that were the nature of the friction, then all investors would immediately adjust their portfolios, or not, optimally. The new market price andexpected return would be immediately established, and remain constant until
the next change in fundamentals. In all of the above examples, however, after
the immediate price response, whose magnitude reflects the size of the shockand the supply of immediately available capital, there is a relatively lengthy period of time over which the price reverts in mean toward its new fundamental
level. In the meantime, additional shocks can occur, with overlapping consequences. The typical pattern suggests that the initial price response is larger
than would occur with perfect capital mobility, and reflects the demand curveof the limited pool of investors that are quickly available to absorb the shock.
The speed of adjustment after the initial price response is a reflection of thetime that it takes more investors to realign their portfolios in light of the newmarket conditions, or for the initially responding investors to gather more capital.
In our setting, as in practice, there can be substantial differences in mean returns across assets that are due not only to cross-sectional differences in “fundamental” cash-flow risks, but are also due to unbalanced distributions of capital, relative to a market without intermediation frictions. Empirical “factor”
models of asset returns do not often account for factors related to the distribution of ownership of assets, or related to likely changes in the distribution ofownership. Exceptions include the recent work of Coval and Stafford (2007)
and Lou (2009), who noted that the conditional mean returns of an equity
tend to be lower due to price pressure when mutual funds owning that equity are experiencing liquidation-motivated outflows, and that the conditionalmean returns recover as price pressure abates. Similarly, Bartram, Griffin, andNg (2010) showed that divergent levels of ownership by national domiciles playa role in equity returns.
In our model, delays in portfolio adjustments are due to the time that it
takes for intermediaries to locate suitable investors. This is only an abstraction, which can also proxy for other forms of delay, including time to educate
investors about assets with which they have limited familiarity (awareness),time for contracting, and time for investors to dispose of their current positions, which could involve similar delays and price shocks, as suggested by
Chaiserote (2008).
Our results extend almost immediately to a model in which, in addition to
obtaining the services of an intermediary, each investor also has a personaltechnology by which opportunities to move capital to the other market arriveat random times, independently across investors, with a constant mean arrival
rate. This would cause only minor modifications to the structure and solutionof our model. Increasing the mean arrival rates of these alternative capitalshifting opportunities reduces the average degree of imbalance of capital and

2496

D. DUFFIE AND B. STRULOVICIthe difference in risk premia between the two markets, and thus reduces the
profitability of intermediation.
Our results also apply with minor alteration to an extension of the model thatallows for randomly timed exit and entrance of investors. For this, investors
would exit at exponentially distributed times that are pairwise independent,
and consume their capital at exit. New investors would appear in proportion
to the current levels of capital. Any difference between exit and entrance rates
would thus be subtracted from the proportional drifts of the capital accumulation processes Xa and Xb .
We could further extend our model so as to treat asymmetric markets. Provided that the local inverse demand function π(·) of each market satisfies similar homogeneity assumptions, intermediation would be characterized by two
distinct thresholds of capital ratios, one for movement of capital from market
a to market b, and another for the reverse movement. For example, if returns
in market a are riskier than those in market b, then, all else equal, capital willbe less mobile toward market a than toward market b. Asymmetry, for example, would allow a consideration of capital mobility from a low-risk “money
market” into a high-risk market such as that for private equity. Many of thequalitative features of our symmetric model, such as the dynamics of capital
mobility and the impact of intermediation competition, are anticipated to carryover to asymmetric settings, at least under regularity conditions.
Another natural extension concerns the case of three or more markets. Consider, for example, three symmetric markets differing only in their capital levels, and satisfying our homogeneity conditions. We conjecture that capital willflow exclusively to the highest-premium market, with more mobility from thelowest-premium market than from the mid-premium market.
APPENDIX A: EQUILIBRIUM ANALYSIS
Here, we provide a stochastic analysis of Markov equilibrium that is more
complete and general than that provided in the main text.
Given an intermediation contact intensity process λ and initial conditions
for capital in each market, we let Xitλ denote the total capital in market i attime t. Given an associated transaction-cost process K, the marginal value to a
supplier of one additional unit of capital in market i at time t is

 ∞
 λ 


λ
−r(s−t)θit = E
(24)e
Ws π XD(s)s ds − Ws− Ks− dNs Ft twhere Ns is the cumulative number of switches back and forth between the twomarkets through time s by the holder of this unit of capital, and the market
indicator D(s) is a or b, depending on whether, at time s, the accumulated
capital Ws is currently located in market a or b. This capital thus accumulates
according to
dWs = Ws− dρD(s−) (s)

CAPITAL MOBILITY AND ASSET PRICING

2497

with initial condition Wt = 1. The market-indicator process D is a marked point
process with an initial condition at time t of D(t) = i, and with an intensity ofjumping from market i to market j at time s of λs 1{Xisλ >Xjsλ } . In the equilibriumthat we shall describe, the value of switching from market i to market j isstrictly positive if and only if Xitλ > Xjtλ  The marginal value of moving capitalis thus λ λ
 λ λ
φλt = max θat θbt − min θat θbt At each time t, intermediaries charge investors some fraction q ∈ [0 1] of
the gain φλt from switching each unit of capital. That is, the proportional intermediation fee is Ktλ = qφλt .
We restrict λ to be a progressively measurable process so that, at each time,the intermediary chooses a contact intensity that depends only on informationthat is currently available.
A monopolistic intermediary’s total rate of fee revenue is λt max(Xat  Xbt ) ×qΦt , where Φt = φλt is the gain from switching capital under policy λ. Givenλλthe initial conditions Xa0= xa and Xb0= xb , and given a gain-from-switchingprocess Φ, the intermediary’s utility for a contact intensity process λ is
 ∞

 λ



−rt
λ
U(xa  xb  Φ λ) = E
e λt qΦt max Xat  Xbt − c dt 0

We assume that the parameters are such that this utility is finite, which is thecase in the equilibria that we analyze. We restrict attention to intermediation
policies that depend on the available information only through the current capital levels (Xat  Xbt ). The intermediary might otherwise prefer to commit once
and for all time to a path-dependent intensity policy that could, at some future
time, be dominated by another policy available at that time, given the current
capital market conditions at that time.
The inability to commit to an intermediation strategy may, in principle, be
overcome by sophisticated punishment threats, as in Ausubel and Deneckere(1989) and Mailath and Samuelson (2006). In such equilibria, if the intermediary deviates, investors would update their beliefs about the intermediary’s
strategy in a way that harms the intermediary. Such equilibria are based on sophisticated off-equilibrium-path investor beliefs, which are not in the spirit of
our assumption that investors are less sophisticated than intermediaries. Another possible justification for our focus on Markov equilibria is the fact that
more sophisticated equilibria unravel in finite-horizon models where (possiblystate-dependent) stage games have a unique Nash equilibrium.
Given the symmetry of the two markets, it suffices to characterize equilibrium behavior in terms of
Xt = max(Xat  Xbt )
Yt = min(Xat  Xbt )

2498

D. DUFFIE AND B. STRULOVICIThe payoff processes to investments in the “larger” and “smaller” markets are,respectively,dρX
t = 1{Xat >Xbt } dρat + 1{Xat ≤Xbt } dρbt dρYt = 1{Xat ≤Xbt } dρat + 1{Xat >Xbt } dρbt 
From the Lévy property, (ρX  ρY ) has the same joint distribution as the primitive payoff processes (ρa  ρb ).
Because we restrict attention to an intermediation intensity process λ that
depends only on current capital levels, and because of symmetry, we can suppose that λt = Λ(Xt  Yt ) for some measurable policy function Λ : R2+ → [0 λ̄]with the property that there is a solution to the associated stochastic differential equation(25)

dXt = −Λ(Xt  Yt )Xt dt + Xt dρX
t 

(26)

dYt = Λ(Xt  Yt )Xt dt + Yt dρYt 

Letting L denote the space of intermediation intensity processes of thisform, and given an assumed gain-from-switching process Φ, the intermediarysolves the problem(27)

sup U(x y Φ λ)
λ∈L

An equilibrium is a pair (λ Φ) consisting of an intermediation intensity process λ that attains the supremum (27) given Φ, and such that Φt = φλt . Thisdefinition includes consistency with the optimality for investors to move theircapital, in exchange for the marginal fee determined by Φ, when contacted by
the intermediary, and includes consistency between the conjectured and actualdynamics for capital movements and search intensity.
A.1. Nonnegativity of the Gain From Switching f
We now prove Proposition 2, allowing for general γ. Because the right-handside of (37) is strictly positive, f or f  must be strictly positive. This implies
that f cannot cross 0 from above. Hence, f must be strictly positive on someinterval of the form (z ∞), and is nonpositive on [1 z] for some level z. It
remains to show that z = 1. Because f (1) = 0, the intermediary does not searchwhen the markets have equal levels of capital, given that c > 0. That is, L(z)vanishes on a neighborhood of 1. From (37), this implies that f is positive onthat neighborhood, which concludes the proof.
The total-present-value conservation equation isV (x y) + xG(x y) + yH(x y) = R(x y) − PT (x y)

CAPITAL MOBILITY AND ASSET PRICING

2499

where R(x y) is the present value of the total future cash flows at rateXt π(Xt ) + Yt π(Yt ), to be divided among the intermediaries and the investors,
and PT (x y) is intermediary’s expected discounted search costs over the infinite horizon, given a trigger T .
Because π is homogeneous of degree −1, we have R(x y) = 2/r. The
search-cost present value PT (1 0) solvesPT (1 0) = p + E e−rτ PT (1 0) 
(28)
where p is present value of search costs from time zero to the exponentiallydistributed time τ of the next loss event. We now show that, for the case of norecovery at loss event,(29)

PT (1 0) =c λ̄ 
1 − e−(2η+r)a(T ) rwhere a(T ) = log(1 + 1/T )/λ̄.
Starting with X0 = 1 and Y0 = 0, for t < τ we havedXt = −λ̄Xt 1{Zt >T } dt
anddYt = λ̄Xt 1{Zt >T } dt
This yields Xt = e−λ̄t and Yt = 1 − e−λ̄t , for t < τ. The intermediary will stop
searching at that time a(T ) at which Za(T ) = T , soe−λ̄a(T )= T
1 − e−λ̄a(T )This yields


1+T
1

a(T ) = logTλ̄The present value of search costs until the next loss event is min(a(T )τ)
p=E


−rt

e
0

λ̄c dt =



λ̄c 
1 − E e−r min(a(T )τ) rBecause τ is exponentially distributed with parameter 2η,
E e−rτ =2η
2η + r

2500

D. DUFFIE AND B. STRULOVICIand 2η + re−r(2η+r)a(T )E e−r min(a(T )τ) =2η + rSubstitution of these into (28) yields the result (29).
A.2. Proof of Proposition 3
That v is bounded follows from the fact that it is dominated by 2/r. The
monotonicity result is based on two intermediate lemmas.
First, given the function f determining intermediation fees, let 1 + ηg0−γ
− f (z)
(z) = 1 − z
r + 2η
The first term of (z) is the present value of switching capital to the undercapitalized market if the intermediary arrests intermediation efforts from thepoint at which the capital ratio Zt is at z until the next loss event occurs, given
g0 . (See (9).) Suppose, in particular, a given reduced policy L(z) = Λ(z 1), anda particular z at which L(z) = 0. Then (z) = 0. As a special case, (1) = 0(which can also be checked directly from the definition of  and the fact thatf (1) = 0). We note that, since the first term defining  is strictly increasing
in z,  (z) must be positive whenever f  (z) is negative. Given a policy L, weshow that  is nonnegative. To see this, we observe that, for z ≥ 1, (37) can be
rewritten as
L(z) (1 − q) + zγ f + z(1 + z)f  = (r + 2η)(z)
(30)
We already know that (1) = 0. Since f is positive from Proposition 2, this implies that f  (z) is negative whenever (z) ≤ 0, and hence that  > 0 whenever ≤ 0. Therefore,  cannot cross 0 from above, which proves our first lemma.
LEMMA 1: For any policy,  is everywhere nonnegative.
This result is intuitive: other things equal, the expected gain from moving
one’s capital is larger if the intermediary immediately stops switching capital
after that last movement, since the difference between capital levels, and hence
between premia, is larger in that case. Lemma 1 has a crucial consequence forthe case γ = 1: The rate at which fees are paid to the intermediary when hesearches is strictly increasing in z. The more heterogeneous the markets, thehigher is the intermediary’s immediate profit from switching. Since this rateof fee payment, net of search costs, is qzf (z) − c, we must show that zf (z) is
strictly increasing in z. We can rewrite (30) when γ = 1 as
L(z)(1 + z) f (z) + zf  (z) = (r + 2η)(z) + qL(z)f (z)

CAPITAL MOBILITY AND ASSET PRICING

2501

Since f is positive and  is nonnegative, this implies that f (z) + zf  (z) is positive whenever L(z) > 0, hence that zf (z) is strictly increasing in z. On any
interval on which L(z) = 0, we have f (z) = (1 + ηg0 )/(r + 2η)(1 − 1/z), so f
is strictly increasing, and, a fortiori, so is zf (z).
LEMMA 2: For γ = 1 and any policy, the revenue rate zf (z) is strictly increasingin z.
We can now show monotonicity of v for any trigger policy. From (13), v isconstant for z ≤ T . Starting with some capital ratio Z0 = z > T ,
 τ



v(z) = Ee−rt qf (Zt )Zt − c 1{Zt >T } dt + e−rτ v0 0where τ is the time of the next loss event. The function z → [qf (z)z − c]1z>T
is nondecreasing in z from Lemma 2, and strictly increasing for z > T . For
T < z < z  , this implies that v(z) < v(z  ) (because the event time τ has a distribution that does not depend on z or z  ). This proves Proposition 3.
A.3. Optimality of a Trigger Policy
This appendix shows that, for any equilibrium pair (f L), the reduced policy
function L must be a trigger policy. In fact, we show that, for any switching-gain
function f that can arise as the result of an admissible intermediation policy,
equilibrium or otherwise, the optimal policy must be of the trigger form.
From Appendix E.2, we know that, for a given f , any bounded solution of
the HJB equation is the value to the intermediary of an optimal policy. We
also know that f is continuous (and, in fact, differentiable) from (10). From
Lemma 2, we also know that, for any admissible policy, zf (z) must be increasing. Finally, f must be such that the value function v is bounded by 2/r. These
conditions define what we call “admissibility” of f . In particular, these conditions must be satisfied in any equilibrium.
We first show that there exists a solution to the HJB equation that is achievedby a trigger policy. Then we verify that any policy that achieves the value function that solves the HJB equation must be of the trigger form.
For any equilibrium, the function f is bounded, because2f (z) = h(z) − g(z) ≤ h(z) + g(z) ≤ h(z) + zg(z) ≤ rTherefore, given any candidates for the capital trigger ratio T and the constantv1 , one can integrate the HJB equation (14) on [T ∞). The smooth-pasting
condition is satisfied if v (T ) = 0, and this is equivalent to the condition thatqTf (T ) = c. (For this, see (11).) Given f , this uniquely determines T , because
Tf (T ) is strictly increasing in T , by Lemma 2. The only difficulty is to show the2502

D. DUFFIE AND B. STRULOVICI

consistency condition v1 = (2η/2η + r)v0 (see (13)), where v0 = limz→∞ v(z),noting that v0 enters as a coefficient of ordinary differential equation (ODE)(14) (in the constant d). To show this, we exploit the linearity of the ODE (14).
Making the change of variables u(z) = v(z) − v1 , we have u(T ) = 0. The dynamics of u do not depend on v0 , in that(31)

u(z) + αz(1 + z)u (z) = β(z)

where β(z) = λ̄(qzf (z) − c)/(r + 2η) and α = λ̄/(r + 2η) > 0 is positive on(T ∞). Moreover, the limit u∞ is, by construction, equal to v0 − v1 . This allows
us to re-express the consistency condition as u∞ = (r/2η + r)v0 . Therefore,
having integrated u over [T ∞), one may simply read off the values v0 and v1 .
The resulting function v(t) = u(t) + v1 solves the initial HJB equation with av0 -dependent coefficient, and also satisfies the smooth-pasting condition.
Thus, for any admissible f , there is an optimal policy of the trigger form.
To conclude, we show that there are no policies solving the HJB equation thatare not of the trigger form. This follows from the linearity in  of the HJB
equation, implying a bang-bang solution, which is strict because indifferenceis characterized by the equation qzf (z) = c, which has a unique solution by
Lemma 2. This analysis is summarized as follows.
PROPOSITION 12: Suppose that the payout-rate function π is of the form
π(x) = k0 + k/x. Then any equilibrium intermediation policy Λ corresponds to a
trigger capital ratio T . That is, Λ(x y) = λ̄1{x/y>T } .
A.4. Existence of Equilibrium
So far, we have shown that any equilibrium must be of the trigger form. In
this appendix, we show that there exists such an equilibrium. Appendix A.5
shows uniqueness of such equilibria.
For any candidate trigger capital ratio T , let f (z|T ) be the net expected gainfrom switching capital across markets under the policy with trigger T , given
current market heterogeneity z. We need to show that there exists some T suchthat qTf (T |T ) = c, that is, such that the intermediary ceases intermediation,
given the switching-gain function f (·) = f (·|T ), exactly when z = T . It suffices
to show that Tf (T |T ) takes all values between 0 and ∞ as T varies from 1 to∞.
Because zf (z) is increasing, Equation (9) implies thatf (z|T ) ≥(T − 1)T (r + 2η)z ≥ T

This implies that Tf (T |T ) ≥ (T − 1)/(r + 2η). We note that the lower boundgrows linearly with T . Because Tf (T |T ) = 0 for T = 1, we know that T →
Tf (T |T ) goes from 0 to ∞ as T goes from 1 to ∞. This function is continuous,
so there exists some T ∗ such that T ∗ f (T ∗ |T ∗ ) = c/q.

CAPITAL MOBILITY AND ASSET PRICING

2503

PROPOSITION 13: Suppose that the payout-rate function π is of the form
π(x) = k0 + k/x. Then, there exists an equilibrium with a trigger policy.
A.5. Trigger Uniqueness
PROOF OF PROPOSITION 4: Suppose that trigger levels S and T , with S < T ,both satisfy the equations of the proposition. Let φ(z) = f T (z) − f S (z) denotethe difference between the gains from switching capital under policies S and T ,as a function of z. (Throughout, we use superscripts to denote dependence on
S or T .) From (16), S < T implies that g0S > g0T . Optimality of S (respectively,
T ) with respect to f S (respectively, f T ) implies that, for any z in (S T ],
 
qzf S (z) − c − z(1 + z) vS (z) > 0and qzf T (z) − c − z(1 + z) vT (z) ≤ 0
Because (vT ) (z) = 0 for z in this interval (S < T ], while (vS ) (z) ≥ 0 by Proposition 3, we know that φ(T ) < 0. Subtracting the version of Equation (7) for T
from the version of the same equation for S yields


1

(32)
 z > T
(a + z)φ + z(1 + z)φ = α 1 −
z
where
a=

r + 2η
+ (1 − q) > 0λ̄α=η(g0T − g0S )
< 0
λ̄

and

Because φ(T ) < 0, this20 implies that φ < 0 for z > T , so that φ is everywhere
negative.
By definition, g0 is the marginal value of capital held by investors in the overcapitalized market, when x = 1 and y = 0 (i.e., when no investor is initiallypresent in the small market). Therefore,
(33)
20

g0 =

2
− Φ0 
r

Indeed, φ(z) = 0 implies that φ (z) < 0, so φ cannot cross zero from below.

2504

D. DUFFIE AND B. STRULOVICI

where Φ0 is the expected discounted value of all future fees that investors
will pay to the intermediary. (Recall that 2/r is the expected discounted
stream of dividends paid on both markets.) We have seen that φ < 0, thatis, f S (z) > f T (z) for all z > T . This means that investors pay, for any z, more
fees with S than with T for z > T . Moreover, for z ∈ [S T ], investors pay fees
(which are positive, from Proposition 2) for trigger S, whereas they pay nothing for trigger T . Therefore, ΦS0 > ΦT0 , which implies from (33) that g0S < g0T , acontradiction.
Q.E.D.
APPENDIX B: CONNECTEDNESSIn this appendix, we outline a model with the natural feature that an investor
is increasingly likely to be in contact with multiple intermediaries at the pointof bargaining as the total number of intermediaries is increased.
Suppose that there is an advertising medium handling intermediary ads. Anintermediary’s effort corresponds to the probability p that its advertisement
will place the intermediary in contact with an investor at the time at which theinvestor checks the medium. We assume that p is bounded by some capacity
constraint p̄ < 1. Each investor, pairwise independently across investors, hassome exogenous intensity χ for the times of monitoring his capital and observing the advertising medium.21 This is consistent with the framework of ourmain model: The intensity of times at which an investor is contacted by at least
one intermediary is χφn (p), whereφn (p) = 1 − (1 − p)n Then, λ̄ = χφn (p̄) is the intermediation capacity parameter of the basic model.
Assuming that a well-connected investor initiates bargaining with a randomly
selected intermediary from among those contacted, each intermediary has
maximal contact intensity λ̄/n. The probability that, when in contact with anintermediary, an investor is in contact with at least two intermediaries isψn (p) = 1 − (1 − p)n − np(1 − p)n−1 For a fixed φ̄ ∈ [0 1], let p̄n solve φn (p̄n ) = φ̄, so that λ̄ is independent of n, as
in our basic model. One may easily check that p̄n is decreasing in n. Moreover,using thatψn (pn ) = φ̄ − npn (1 − pn )n−1 = φ̄ −21

(1 − φ̄)npn

1 − pnAt such times, the investor observes the medium and plays a bargaining game with advertised
intermediaries. If bargaining breaks down, the investor leaves his capital in the large market, until
the next monitoring time.

CAPITAL MOBILITY AND ASSET PRICING

2505

one can show that ψn (pn ) is increasing in n.22 Therefore, keeping constantthe flow of investors being contacted at any given time, the average numberof intermediaries in contact with any given investor is increasing in n. As thenumber of intermediaries goes to infinity, the probability that investor is wellconnected islim ψn (pn ) = φ̄ + (1 − φ̄) log(1 − φ̄)n→∞The second term is negative. This specification can be generalized to an arbitrary number of media, with the same result that ψn is increasing in n.
APPENDIX C: PROOFS OF RESULTS IN SECTION 4.3
PROOF OF PROPOSITION 10: As before, we let g0L = G(1 0), under strategy
L. For any equilibrium with aggregate mobility z → L(z) and fee z → q(z),one can easily modify the proofs of Lemmas 1 and 2 to show that
11 + ηg0LL (z) ≡
1−
− f L (z)r + 2η
z
is nonnegative and that zf L (z) is strictly increasing in z. If T̄ ≤ T1 , we have

1
1 + ηg01
1
1−
f (T̄ ) =r + 2η
T̄
where f 1 and g01 denote the corresponding quantities for the monopolistic case,since the intermediary does not search at T̄ . Further,
1
1 + ηg0LL1−
f (T̄ ) ≤r + 2η
T̄
from the nonnegativity of L (T̄ ). Therefore,

 1

 1
g0 − g0LL(34)(T̄ − 1)T̄ f (T̄ ) − f (T̄ ) ≥ η
r + 2η
Since g0L ≤ 2/r for any policy, there exists, for any ε > 0, some η̄ such that,for all η < η̄, the right-hand side of (34) is bounded in norm by ε whenever22
To verify this, one is to show that npn /(1 − pn ) is decreasing. Expressing pn in terms of
α = (1 − φ̄)−1 > 1 and letting x = 1/n, this is equivalent to showing that (αx − 1)/x is increasing
in x. This is easily done by checking the positivity of the derivative, whose numerator is increasing
in u = αx and vanishes for u = 1.

2506

D. DUFFIE AND B. STRULOVICIT̄ ≤ T1 , since we also have an upper bound on T1 from (16). Choosing ε below(1/q(n) − 1/q(1))c and setting η̄ accordingly, we have, for any T̄ ≤ T1 ,(35)

q(1)T̄f 1 (T̄ ) ≥ q(1) 
q(1) 
q(n)T̄f L (T̄ ) − q(n)ε ≥c − q(n)ε > c
q(n)
q(n)

which shows that it is strictly optimal for the monopolist to search at T̄ , conQ.E.D.
tradicting the assumption that T̄ ≤ T1 .
PROOF OF PROPOSITION 11: At Sn , it cannot be strictly profitable for anintermediary to deviate by continuing to search and receive the net payoff
q(1)Sn f L (Sn ) − c per unit of effort, but it was profitable to some intermediaries to search at a capital heterogeneity just above Sn . This implies that Sn
must satisfy the equation
q(1)Sn f L (Sn ) = cIn words, there is a single active intermediary just before Sn is reached. We
recall from the monopolistic case that T1 satisfies the equation
q(1)T1 f 1 (T1 ) = cTherefore, it suffices to show that the roots of these two equations are arbitrarily close if η is arbitrarily small. We have

1
1 + ηg0LL
1−
f (Sn ) =r + 2η
Snand

1 + ηg01
f (T1 ) =r + 2η
1



1

1−
T1

Therefore, Sn and T1 must satisfy
1 + ηg01
1 + ηg0L(Sn − 1) −(T1 − 1) = 0r + 2η
r + 2η
which may be rewritten as


 L

g0 − g01
1 + ηg0L(Sn − T1 ) = η(1 + T1 )
r + 2η
r + 2η
Because T1 is uniformly bounded from (16) and because both g0L and g01 arebounded by 2/r, the right-hand side is less than ε if η is chosen small enough.
The first factor of the left-hand side is equivalent to 1/r when η is small

CAPITAL MOBILITY AND ASSET PRICING

2507

enough. Combining these observations shows that |Sn − T1 | ≤ ε for any arbitrary ε > 0, provided that η is small enough.
Q.E.D.
Appendices D through L are located in Duffie and Strulovici (2012), theSupplemental Material to this paper.
REFERENCESAMERIKS, J., AND S. ZELDES (2004): “How Do Household Portfolio Shares Vary With Age?”
Working Paper, Columbia University. [2493]
ANDRADE, S., C. CHANG, AND M. SEASHOLES (2008): “Trading Imbalances, Predictable Reversals, and Cross-Stock Price Pressure,” Journal of Financial Economics, 88, 406–423. [2494]
AUSUBEL, L. M., AND R. J. DENECKERE (1989): “Reputation in Bargaining and Durable Goods
Monopoly,” Econometrica, 57, 511–531. [2497]
BARTRAM, S. M., J. GRIFFIN, AND D. NG (2010): “How Important Is Foreign Ownership forInternational Stock Co-Movement?” Working Paper, Wharton School. [2495]
BASAK, S., AND B. CROITORU (2000): “Equilibrium Mispricing in a Capital Market With Portfolio
Constraints,” Review of Financial Studies, 13, 715–748. [2470]
BHATTACHARYA, S., AND K. M. HAGERTY (1987): “Dealerships, Trading Externalities, and General Equilibrium,” in Contractual Arrangements for Intertemporal Trade. Minnesota Studies inMacroeconomics Series, Vol. 1, ed. by E. Prescott and N. Wallace. Minneapolis: University of
Minnesota Press, 81–104. [2470]
BILIAS, Y., D. GEORGARAKOS, AND M. HALIASSOS (2010): “Portfolio Inertia and Stock Market
Fluctuations,” Journal of Money, Credit, and Banking, 42, 715–742. [2494]
BORN, P., AND K. VISCUSI (2006): “The Catastrophic Effects of Natural Disasters on InsuranceMarkets,” Journal of Risk and Uncertainty, 33, 55–72. [2493]
BRUNNERMEIER, M., AND S. NAGEL (2008): “Do Wealth Fluctuations Generate Time-Varying
Risk Aversion? Micro-Evidence on Individuals’ Asset Allocation,” American Economic Review,
98, 713–736. [2494]
CHAISEROTE, S. (2008): “Liquidity Provision by International Institutional Investors: Evidence
From Brazilian Privatization,” Working Paper, Stanford University. [2495]
CHAN, W. S. (2003): “Stock Price Reaction to News and No-News: Drift and Reversal After
Headlines,” Journal of Financial Economics, 70, 223–260. [2494]
CHEN, H., G. NORONHA, AND V. SINGHAL (2004): “The Price Response to S&P 500 Index Additions and Deletions: Evidence of Asymmetry and a New Explanation,” Journal of Finance, 59,
1901–1929. [2494]
CHEN, Z., A. LOOKMAN, AND N. SCHÜRHOFF (2008): “Why Ratings Matter: Evidence From
Lehman’s Index Inclusion Rule Change,” Working Paper, University of Lausanne. [2494]
COHEN, L., AND D. LOU (2012): “Complicated Firms,” Journal of Financial Economics, 104 (2),383–400. [2494]
COVAL, J., AND E. STAFFORD (2007): “Asset Fire Sales (and Purchases) in Equity Markets,” Journal of Financial Economics, 86, 479–512. [2494,2495]
DELLAVIGNA, S., AND J. M. POLLET (2009): “Investor Inattention and Friday Earnings Announcements,” Journal of Finance, 64, 709–749. [2494]
DUFFIE, D., AND B. STRULOVICI (2012): “Supplement to ‘Capital Mobility and Asset
Pricing’,” Econometrica Supplemental Material, 80, http://www.econometricsociety.org/ecta/
Supmat/8822_extensions.pdf. [2471,2507]
DUFFIE, D., N. GÂRLEANU, AND L. H. PEDERSEN (2005): “Over-the-Counter Markets,” Econometrica, 73, 1815–1847. [2470](2007): “Valuation in Over-the-Counter Markets,” Review of Financial Studies, 20,
1865–1900. [2470]

2508

D. DUFFIE AND B. STRULOVICIENGELBERG, J. E., A. V. REED, AND M. C. RINGGENBERG (2012): “How Are Shorts Informed?
Short Sellers, News, and Information Processing,” Journal of Financial Economics, 105 (2),260–278. [2494]
ENZ, R. (2001): The Insurance Cycle as an Entrepreneurial Challenge. Zurich, Switzerland: Swiss
Re Technical Publishing. [2493]
FELDHÜTTER, P. (2009): “The Same Bond at Different Prices: Identifying Search Frictions andSelling Pressures,” Review of Financial Studies, 25, 1155–1206. [2494]
FROOT, K. (2001): “The Market for Catastrophe Risk: A Clinical Examination,” Journal of Financial Economics, 60, 529–571. [2470]
FROOT, K., AND P. O’CONNELL (1999): “The Pricing of US Catastrophe Reinsurance,” in The Financing of Catastrophe Risk, ed. by K. Froot. Chicago, IL: University of Chicago Press, 195–232.
[2470,2493]
GEHRIG, T. (1993): “Intermediation in Search Markets,” Journal of Economics and Management
Strategy, 2, 97–120. [2470]
GREEN, R. (2007): “Issuers, Underwriter Syndicates, and Aftermarket Transparency,” Journal of
Finance, 62, 1529–1550. [2487]
GREENWOOD, R. (2005): “Short and Long Term Demand Curves for Stocks: Theory and Evidence,” Journal of Financial Economics, 75, 607–650. [2494]
GROMB, D., AND D. VAYANOS (2012): “Financially Constrained Arbitrage and Cross-Market
Contagion,” Working Paper, London School of Economics. [2470]
HARRIS, L., AND E. GUREL (1986): “Price and Volume Effects Associated With Changes in the
S&P 500: New Evidence of Price Pressures,” Journal of Finance, 41, 815–830. [2494]
HE, Z., AND A. KRISHNAMURTHY (2012): “A Model of Capital and Crises,” Review of Economic
Studies, 79 (2), 735–777. [2470]
HOLTHAUSEN, R. W., R. LEFTWICH, AND D. MAYERS (1990): “Large-Block Transactions, theSpeed of Response and Temporary and Permanent Stock-Price Effects,” Journal of Financial
Economics, 26, 71–95. [2494]
HRADSKY, G. T., AND R. D. LONG (1989): “High-Yield Default Losses and the Return Performance of Bankrupt Debt,” Financial Analysts Journal, 45, 38–49. [2494]
KAUL, A., V. MEHROTRA, AND R. MORCK (2000): “Demand Curves for Stocks do Slope Down:
New Evidence From an Index Weights Adjustment,” Journal of Finance, 55, 893–912. [2494]
KULAK, J. P. (2008): “Price-Pressures and Foreign Investor Biases Revealed Through Firms: Geographical Placement Choices,” Working Paper, University of Lausanne. [2494]
LAGOS, R., G. ROCHETEAU, AND P. WEILL (2011): “Crises and Liquidity in Over-the-Counter
Markets,” Journal of Economic Theory, 146, 2169–2205. [2470]
LOU, D. (2009): “A Flow-Based Explanation for Return Predictability,” Working Paper, Yale
University, Review of Financial Studies (forthcoming). [2495]
LUSARDI, A. (1999): “Information, Expectations, and Savings,” in Behavioral Dimensions of Retirement Economics, ed. by H. Aaron. Washington, DC: Brookings Institution Press, 81–115.
[2494](2003): “Planning and Savings for Retirement,” Unpublished Manuscript, DartmouthCollege. [2494]
MAILATH, G., AND L. SAMUELSON (2006): Repeated Games and Reputations: Long-Run Relationships. Oxford, U.K.: Oxford University Press. [2497]
MITCHELL, M., L. H. PEDERSEN, AND T. PULVINO (2007): “Slow Moving Capital,” American
Economic Review, Papers and Proceedings, 97, 215–220. [2494]
MITCHELL, O. S., G. R. MOTTOLA, S. P. UTKUS, AND T. YAMAGUCHI (2006): “The Inattentive
Participant: Portfolio Trading Behavior in 401(k) Plans,” Working Paper, University of Pennsylvania. [2493]
MORESI, S. (1991): “Three Essays in Economic Theory,” Ph.D. Thesis, MIT. [2470]
MOU, Y. (2011): “Limits to Arbitrage and Commodity Index Investment: Front-Running theGoldman Roll,” Working Paper, Columbia University. [2494]

CAPITAL MOBILITY AND ASSET PRICING

2509

NEWMAN, Y., AND M. RIERSON (2003): “Illiquidity Spillovers: Theory and Evidence From European Telecom Bond Issuance,” Working Paper, Graduate School of Business, Stanford University. [2494]
PROTTER, P. (2004): Stochastic Integration and Differential Equations (Second Ed.). New York:
Springer-Verlag. [2472]
QUAH, J., AND B. STRULOVICI (2009): “Comparative Statics, Informativeness, and the Interval
Dominance Order,” Econometrica, 77, 1949–1992. [2483](2012): “Discounting, Values, and Decisions,” Working Paper, Northwestern University.
[2483]RUBINSTEIN, A. (1982): “Perfect Equilibrium in a Bargaining Model,” Econometrica, 50, 97–108.
[2488]RUBINSTEIN, A., AND A. WOLINSKY (1987): “Middlemen,” Quarterly Journal of Economics, 102,581–594. [2470]
SCHOLES, M. (1972): “The Market for Securities: Substitution versus Price Pressure and the Effects of Information on Share Prices,” Journal of Business, 45, 179–211. [2494]
SHLEIFER, A. (1986): “Do Demand Curves for Stocks Slope Down?” Journal of Finance, 41,
579–590. [2494]
STOLE, L., AND J. ZWIEBEL (1996): “Intra-Firm Bargaining Under Non-Binding Contracts,” Review of Economic Studies, 63, 375–410. [2488]
SUN, Y. (2006): “The Exact Law of Large Numbers via Fubini Extension and Characterization of
Insurable Risks,” Journal of Economic Theory, 126, 31–69. [2475]
SUTTON, J. (1986): “Non-Cooperative Bargaining Theory: An Introduction,” Review of Economic
Studies, 53, 709–724. [2487]
VAYANOS, D., AND T. WANG (2007): “Search and Endogenous Concentration of Liquidity in Asset
Markets,” Journal of Economic Theory, 136, 66–104. [2470]
WEILL, P.-O. (2007): “Leaning Against the Wind,” Review of Economic Studies, 74, 1329–1354.
[2475](2008): “The Liquidity Premium in a Dynamic Bargaining Market,” Journal of Economic
Theory, 140, 66–96. [2470]
YAVAŞ, A. (1996): “Search and Trading in Intermediated Markets,” Journal of Economics andManagement Strategy, 5, 195–216. [2470]
ZANJANI, G. (2002): “Pricing and Capital Allocation in Catastrophe Insurance,” Journal of Financial Economics, 65, 283–305. [2493]

Graduate School of Business, Stanford University, Stanford, CA 94305, U.S.A.;
duffie@stanford.edu
andDept. of Economics, Northwestern University, Evanston, IL 60208, U.S.A.;
b-strulovici@northwestern.edu.
Manuscript received September, 2009; final revision received January, 2012.